The goal of the proposal is to study efficient mixed finite element methods for the computation of transport maps in optimal transportation problems. The focus is on problems in which the cost of transport is a quadratic function of the distance. They lead to Monge-Ampere equations. The first part of the project consists in clarifying the applicability of finite element type methods to weak solutions of the equation. The key approach here is approximation by smooth functions. In the second part, techniques of mixed finite element analysis are adapted to the approximation of smooth solutions of the equation.
Optimal transportation has a growing application in various fields ranging from theoretical ones such as geometry and analysis to applied fields such as biology, pattern recognition, image processing, fluid mechanics, geophysics, meteorology, optics, oceanography and cosmology. This has created the critical need for efficient and robust numerical methods backed up theoretically to solve optimal transportation problems. The efficient and reliable methods developed from this project could be used to solve optimal transportation problems which appear in many other applications e.g. weather forecasting, traffic congestion, economics, mesh equidistribution, texture mapping, etc. The proposal studies the Monge-Ampere equation of optimal transportation with the goal of clarifying theoretically the use of the efficient mixed finite element methods. It advances knowledge towards the resolution of some open problems in analysis and geometry involving Monge-Ampere type equations.
